Model specification
In addition to the number of profiles (specified with the n_profiles argument), the model can be specified in terms of whether and how the variable variances and covariances are estimated.
The models are specified by passing arguments to the variance and covariance arguments. The possible values for these arguments are:
variances: “equal” and “varying”covariances: “varying”, “equal”, and “zero”
If no values are specified for these, then the variances are constrained to be equal across classes, and covariances are fixed to 0 (conditional independence of the indicators).
These arguments allow for four models to be specified:
- Equal variances and covariances fixed to 0 (Model 1)
- Varying variances and covariances fixed to 0 (Model 2)
- Equal variances and equal covariances (Model 3)
- Varying variances and varying covariances (Model 6)
Two additional models (Models 4 and 5) can be fit using MPlus. More information on the models can be found in the vignette.
Here is an example of specifying a model with varying variances and covariances (Model 6):
pisaUSA15[1:100, ] %>%
select(broad_interest, enjoyment, self_efficacy) %>%
estimate_profiles(3,
variances = "varying",
covariances = "varying")
#> Warning in estimate_profiles_mclust(df_full, n_profiles, model_numbers, : The mclust algorithm does not allow for missing data. Some rows were omitted from analysis. Consider using Mplus, which accounts for cases with partially missing data, or use a non-parametric single imputation technique prior to analysis, such as the R-package 'missForest'.
#> tidyLPA analysis using mclust:
#>
#> Model Classes AIC BIC Entropy prob_min prob_max n_min n_max BLRT_p
#> 6 3 583.13 656.89 0.86 0.87 0.98 0.16 0.62 0.04
In general, the approach to choosing the model is similar to choosing the number of profiles, requiring deciding on the basis of evidence from multiple sources, including information criteria, statistical tests, and concerns of interpretability and parsimony. The article by Pastor and colleagues (2007) has helpful information on the model specifications. Here, the six models that are possible to specify in LPA are described in terms of how the variables used to create the profiles are estimated.
Note that p represents different profiles and each parameterization is represented by a 4 x 4 covariance matrix and therefore would represent the parameterization for a four-profile solution. In all of the models, the means are estimated freely in the different profiles. Imagine that each row and column represents a different variable, i.e., the first row (and column) represents broad interest, the second enjoyment, the third self-efficacy, and the fourth another variable, i.e., future goals and plans.
1. Equal variances, and covariances fixed to 0 (model 1)
In this model, which corresponds to the mclust model wit the name “EEI”, the variances are estimated to be equal across profiles, indicated by the absence of a p subscript for any of the diagonal elements of the matrix. The covariances are constrained to be zero, as indicated by the 0’s between every combination of the variables.
It is specified with variances = "equal" and covariances = "zero".
This model is highly constrained but also parsimonious: the profiles are estimated in such a way that the variables’ variances are identical for each of the profiles, and the relationships between the variables are not estimated. In this way, less degrees of freedom are taken used to explain the observations that make up the data. However, estimating more parameters–as in the other models–may better explain the data, justifying the addition in complexity that their addition involves (and their reduction in degrees of freedom). This model is sometimes referred to as a class-invariant parameterization.
\[
\left[ \begin{matrix} { \sigma }_{ 1 }^{ 2 } & 0 & 0 & 0 \\ 0 & { \sigma }_{ 2 }^{ 2 } & 0 & 0 \\ 0 & 0 & { \sigma }_{ 3 }^{ 2 } & 0 \\ 0 & 0 & 0 & { \sigma }_{ 4 }^{ 2 } \end{matrix} \right]
\]
2. Varying variances and covariances fixed to 0 (model 2)
This model corresponds to the mclust model “VVI” and allows for the variances to be freely estimated across profiles. The covariances are constrained to zero.
It is specified with variances = "varying" and covariances = "zero".
Thus, it is more flexible (and less parsimonious) than model 1, but in terms of the covariances, is more constrained than model 2. This model is sometimes referred to as a class-varying diagonal parameterization.
\[
\left[ \begin{matrix} { \sigma }_{ 1p }^{ 2 } & 0 & 0 & 0 \\ 0 & { \sigma }_{ 2p }^{ 2 } & 0 & 0 \\ 0 & 0 & { \sigma }_{ 3p }^{ 2 } & 0 \\ 0 & 0 & 0 & { \sigma }_{ 4p }^{ 2 } \end{matrix} \right]
\]
3. Equal variances and equal covariances (model 3)
This model corresponds to the mclust model “EEE”. In this model, the variances are still constrained to be the same across the profiles, although now the covariances are estimated (but like the variances, are constrained to be the same across profiles).
It is specified with variances = "equal" and covariances = "equal".
Thus, this model is the first to estimate the covariance (or correlations) of the variables used to create the profiles, thus adding more information that can be used to better understand the characteristics of the profiles (and, potentially, better explain the data). This model is sometimes referred to as a class-invariant unrestricted parameterization.
\[
\left[ \begin{matrix} { \sigma }_{ 1 }^{ 2 } & { \sigma }_{ 21 } & { \sigma }_{ 31 } & { \sigma }_{ 41 } \\ { \sigma }_{ 12 } & { \sigma }_{ 2 }^{ 2 } & { \sigma }_{ 23 } & { \sigma }_{ 24 } \\ { \sigma }_{ 13 } & { \sigma }_{ 12 } & { \sigma }_{ 3 }^{ 2 } & { \sigma }_{ 33 } \\ { \sigma }_{ 14 } & { \sigma }_{ 12 } & { \sigma }_{ 12 } & { \sigma }_{ 4 }^{ 2 } \end{matrix} \right]
\]
4. Varying means, varying variances, and equal covariances (model 4)
This model, which specifies for the variances to be freely estimated across the profiles and for the covariances to be estimated to be equal across profiles, extends model 3.
It is specified with variances = "varying" and covariances = "equal".
Unfortunately, this model cannot be specified with mclust, though it can be with MPlus; this model can be used with the functions to interface to MPlus described below.
\[
\left[ \begin{matrix} { \sigma }_{ 1p }^{ 2 } & { \sigma }_{ 21 } & { \sigma }_{ 31 } & { \sigma }_{ 41 } \\ { \sigma }_{ 12 } & { \sigma }_{ 2p }^{ 2 } & { \sigma }_{ 23 } & { \sigma }_{ 24 } \\ { \sigma }_{ 13 } & { \sigma }_{ 12 } & { \sigma }_{ 3p }^{ 2 } & { \sigma }_{ 33 } \\ { \sigma }_{ 14 } & { \sigma }_{ 12 } & { \sigma }_{ 12 } & { \sigma }_{ 4p }^{ 2 } \end{matrix} \right]
\]
5. Varying means, equal variances, and varying covariances (model 5)
This model specifies the variances to be equal across the profiles, but allows the covariances to be freely estimated across the profiles.
It is specified with variances = "equal" and covariances = "varying".
Like model 4, this model cannot be specified with mclust, though it can be with MPlus. Again, this model can be used with the functions to interface to MPlus described below.
\[
\left[ \begin{matrix} { \sigma }_{ 1 }^{ 2 } & { \sigma }_{ 21p } & { \sigma }_{ 31p } & { \sigma }_{ 41p } \\ { \sigma }_{ 12p } & { \sigma }_{ 2 }^{ 2 } & { \sigma }_{ 23p } & { \sigma }_{ 24p } \\ { \sigma }_{ 13p } & { \sigma }_{ 12p } & { \sigma }_{ 3 }^{ 2 } & { \sigma }_{ 33p } \\ { \sigma }_{ 14p } & { \sigma }_{ 12p } & { \sigma }_{ 12p } & { \sigma }_{ 4 }^{ 2 } \end{matrix} \right] \quad
\]
6. Varying variances and varying covariances (model 6)
This model corresponds to the mclust model “VVV”. It allows the variances and the covariances to be freely estimated across profiles.
It is specified with variances = "varying" and covariances = "varying".
Thus, it is the most complex model, with the potential to allow for understanding many aspects of the variables that are used to estimate the profiles and how they are related. However, it is less parsimonious than all of the other models, and the added parameters should be considered in light of how preferred this model is relative to those with more simple specifications. This model is sometimes referred to as a class-varying unrestricted parameterization.
\[
\left[ \begin{matrix} { \sigma }_{ 1p }^{ 2 } & { \sigma }_{ 21p } & { \sigma }_{ 31p } & { \sigma }_{ 41p } \\ { \sigma }_{ 12p } & { \sigma }_{ 2p }^{ 2 } & { \sigma }_{ 23p } & { \sigma }_{ 24p } \\ { \sigma }_{ 13p } & { \sigma }_{ 12p } & { \sigma }_{ 3p }^{ 2 } & { \sigma }_{ 33p } \\ { \sigma }_{ 14p } & { \sigma }_{ 12p } & { \sigma }_{ 12p } & { \sigma }_{ 4p }^{ 2 } \end{matrix} \right]
\]
Getting data
Since we often wish to use the estimated profiles in subsequent analyses, we may want the original data.frame, with variables that are predictors or outcomes of the profiles, included. Here, we created profiles with just two of the three variables, to demonstrate how the third variable is still returned in the output. We can return this data.frame, and not just one with the variables used to create the profiles and the profile assignments (and posterior probabilities), using the function get_data():
get_data(m3)
#> # A tibble: 100 × 9
#> model_number classes_number broad_interest enjoyment self_efficacy CPROB1
#> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 1 3 3.8 4 1 1.00e+0
#> 2 1 3 3 3 2.75 8.29e-2
#> 3 1 3 1.8 2.8 3.38 8.73e-4
#> 4 1 3 1.4 1 2.75 8.31e-8
#> 5 1 3 1.8 2.2 2 2.69e-3
#> 6 1 3 1.6 1.6 1.88 1.88e-4
#> 7 1 3 3 3.8 2.25 9.27e-1
#> 8 1 3 2.6 2.2 2 1.03e-2
#> 9 1 3 1 2.8 2.62 1.81e-3
#> 10 1 3 2.2 2 1.75 4.32e-3
#> # … with 90 more rows, and 3 more variables: CPROB2 <dbl>, CPROB3 <dbl>,
#> # Class <dbl>
We note that if more than one model is fit at once, then the data is returned not in the (wide) format above, but in long form, e.g.:
m4 <- pisaUSA15[1:100, ] %>%
select(broad_interest, enjoyment, self_efficacy) %>%
single_imputation() %>%
estimate_profiles(c(3, 4))
get_data(m4)
#> # A tibble: 700 × 9
#> model_number classes_number broad_interest enjoyment self_efficacy Class
#> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 1 3 3.8 4 1 1
#> 2 1 3 3 3 2.75 3
#> 3 1 3 1.8 2.8 3.38 3
#> 4 1 3 1.4 1 2.75 2
#> 5 1 3 1.8 2.2 2 3
#> 6 1 3 1.6 1.6 1.88 3
#> 7 1 3 3 3.8 2.25 1
#> 8 1 3 2.6 2.2 2 3
#> 9 1 3 1 2.8 2.62 3
#> 10 1 3 2.2 2 1.75 3
#> # … with 690 more rows, and 3 more variables: Class_prob <int>,
#> # Probability <dbl>, id <int>
Getting fit statistics
If we wish to work with the fit statistics more programatically, we can use the get_fit() function:
get_fit(m4)
#> # A tibble: 2 × 18
#> Model Classes LogLik AIC AWE BIC CAIC CLC KIC SABIC ICL Entropy
#> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 1 3 -303. 635. 776. 671. 685. 608. 652. 627. -700. 0.758
#> 2 1 4 -292. 620. 802. 667. 685. 586. 641. 610. -681. 0.898
#> # … with 6 more variables: prob_min <dbl>, prob_max <dbl>, n_min <dbl>,
#> # n_max <dbl>, BLRT_val <dbl>, BLRT_p <dbl>
Descriptions of these fit indices follows:
- LogLik: Log-likelihood of the data, given the model.
- AIC: Aikake information criterion; based on -2 log-likelihood, and penalized by number of parameters.
- AWE: Approximate weight of evidence; combines information on model fit and on classification errors (Celeux et al., 1997).
- BIC: Bayesian information criterion; based on -2 log-likelihood, and penalized by number of parameters adjusted by sample size.
- CAIC: Consistent Aikake information criterion; based on -2 log-likelihood, and penalized by number of parameters adjusted by sample size.
- CLC: Classification Likelihood Criterion; based on -2 log-likelihood, and penalized by the entropy (Biernacki, 1997).
- KIC: Kullback information criterion; based on -2 log-likelihood, and penalized by 3 times the number of parameters -1 (Cavanaugh, 1999).
- SABIC: Sample size-adjusted Bayesian information criterion (Sclove, 1987).
- ICL: Integrated completed likelihood (Biernacki, Celeux, & Govaert, 2000).
- Entropy: A measure of classification uncertainty, reverse-coded so that 1 reflects complete certainty of classification, and 0 complete uncertainty (see Celeux & Soromenho, 1996).
- Prob. Min.: Minimum of the diagonal of the average latent class probabilities for most likely class membership, by assigned class. The minimum should be as high as possible, reflecting greater classification certainty (cases are assigned to classes they have a high probability of belonging to; see Jung & Wickrama, 2008).
- Prob. Max.: Maximum of the diagonal of the average latent class probabilities for most likely class membership, by assigned class. The maximum should also be as high as possible, reflecting greater classification certainty (cases are assigned to classes they have a high probability of belonging to).
- N Min.: Proportion of the sample assigned to the smallest class (based on most likely class membership).
- N Max.: Proportion of the sample assigned to the largest class (based on most likely class membership).
- BLRT: bootstrapped likelihood test.
- BLRT p-value: p-value for the bootstrapped likelihood ratio test.
Notes
This is related to prcr, for use of two-step cluster analysis to carry out person-oriented analyses.
To contribute, file issues via GitHub here or get in touch via email or Twitter.
References
Pastor, D. A., Barron, K. E., Miller, B. J., & Davis, S. L. (2007). A latent profile analysis of college students’ achievement goal orientation. Contemporary Educational Psychology, 32(1), 8-47. (https://www.sciencedirect.com/science/article/pii/S0361476X06000543
Helpful resources
Hennig et al’s (2015) handbook for an overview of mixture models, of which LPA is often considered an instance of.
Collins and Lanza (2013) for a book on the related approach (for use with dichotomous, rather than continuous variables used to create the profiles) Latent Class Analysis (LCA)
How to cite tidyLPA
Rosenberg, J. M., Beymer, P. N., Anderson, D. J., Van Lissa, C. J., & Schmidt, J. A. (2018). tidyLPA: An R Package to Easily Carry Out Latent Profile Analysis (LPA) Using Open-Source or Commercial Software. Journal of Open Source Software, 3(30), 978, https://doi.org/10.21105/joss.00978
You can also cite the most latest version with the following citation:
Rosenberg, J. M., van Lissa, C. J., Beymer, P. N., Anderson, D. J., Schell, M. J. & Schmidt, J. A. (2019). tidyLPA: Easily carry out Latent Profile Analysis (LPA) using open-source or commercial software [R package]. https://data-edu.github.io/tidyLPA/
